Normal Distributions

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Chapter 13

• Normal Distributions

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Thought Question 1
The heights of adult women in the United States
follow, at least approximately, a bell-shaped
curve. What do you think that means?
Thought Question 2 What does it mean to
say that a mans weight is in the 30th percentile
for all adult males?
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Thought Question 3
A standardized score is simply the number of
standard deviations an individual falls above or
below the mean for the whole group. (Values
above the mean have positive standardized scores,
while those below the mean have negative ones).
Males (ages 18-24) have a mean height of 70
inches and a standard deviation of 2.8 in.
Females (ages 18-24) have a mean height of 65 in.
and a standard deviation of 2.5 in. Thus a man
who is 72.8 inches tall has a standardized score
of 1. What is the standardized score
corresponding to your height?
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Thought Question 4
Data sets consisting of physical measurements
(heights, weights, lengths of bones, and so on)
for adults of the same species and sex tend to
follow a similar pattern. The pattern is that
most individuals are clumped around the average,
with numbers decreasing the farther values are
from the average in either direction. Describe
what shape a histogram of such measurements would
have.
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Bell-Shaped CurveThe Normal Distributionof
Population Values
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Asymmetric Distributionsof the Population Values
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The Normal Distribution
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With the Mean and Standard Deviation of the
Normal Distribution We Can Determine

• What proportion of individuals fall into any
  range of values
• At what percentile a given individual falls, if
  you know their value
• What value corresponds to a given percentile

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Empirical Rule forAny Normal Curve

• 68 of the values fall within one standard
  deviation of the mean
• 95 of the values fall within two standard
  deviations of the mean
• 99.7 of the values fall within three standard
  deviations of the mean
• 68-95-99.7 Rule

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Empirical Rule forAny Normal Curve
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Health and Nutrition Examination Study of
1976-1980(HANES)

• Heights of adults, aged 18-24
• women
• mean 65.0 inches
• standard deviation 2.5 inches
• men
• mean 70.0 inches
• standard deviation 2.8 inches

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Health and Nutrition Examination Study of
1976-1980(HANES)

• Empirical Rule
• women
• 68 are between 62.5 and 67.5 inches
• mean ? 1 std dev 65.0 ? 2.5
• 95 are between 60.0 and 70.0 inches
• 99.7 are between 57.5 and 72.5 inches
• men
• 68 are between 67.2 and 72.8 inches
• 95 are between 64.4 and 75.6 inches
• 99.7 are between 61.6 and 78.4 inches

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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 72.8 inches
  tall?

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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 68 inches
  tall?

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Standardized Scores

• How many standard deviations is 68 from 70?
• standardized score
• (observed value minus mean) / (std dev)
• (68 - 70) / 2.8 -0.71
• The value 68 is 0.71 standard deviations below
  the mean 70.

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Standardized Scores

• standardized score
• (observed value minus mean) / (std dev)
• z is the standardized score
• x is the observed value
• m is the population mean
• s is the population standard deviation

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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 68 inches
  tall?

-0.71 0 (standardized values)
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Table B Percentiles of the Standardized Normal
Distribution

• See pg. 185 (pg. 547 in text) for Table B.
  (the Standard Normal Table)
• Look up the closest standardized score in the
  table.
• Find the percentile corresponding to the
  standardized score (this is the percent of values
  below the corresponding standardized score or
  z-value).

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Table B Percentiles of the Standardized Normal
Distribution
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What proportion of men are less than 68 inches
  tall?

24.2
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What height value is the 10th percentile for men
  aged 18 to 24?

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Table B Percentiles of the Standardized Normal
Distribution

• See pg. 185 (pg. 547 in text) for Table B.
• Look up the closest percentile in the table.
• Find the corresponding standardized score.
• The value you seek is that many standard
  deviations from the mean.

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Table B Percentiles of the Standardized Normal
Distribution
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Health and Nutrition Examination Study of
1976-1980(HANES)

• What height value is the 10th percentile for men
  aged 18 to 24?

-1.3 0 (standardized values)
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Observed Value for a Standardized Score

• What height value is the 10th percentile for men
  aged 18 to 24?
• observed value
• mean plus (standardized score) ? (std dev)
• 70 (-1.3 ) ? (2.8)
• 70 (?3.64) 66.36
• The value 66.36 is approximately the 10th
  percentile of the population.

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Observed Value for a Standardized Score

• observed value
• mean plus (standardized score) ? (std dev)
• x is the observed value
• m is the population mean
• z is the standardized score
• s is the population standard deviation

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Key Concepts

• Population values are distributed with differing
  shapes, some normal, some non-normal.
• Empirical Rule (68-95-99.7 Rule)
• Standardized Score
• Percentile
• Standard Normal Table